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Frank Morgan's Math Chat - Burning 1-Hour Fuses in 45 Minutes

February 4, 1999

BURNING 1-HOUR FUSES IN 45 MINUTES

OLD CHALLENGE. You have two one-hour fuses: lighting one end of a fuse will cause it to burn down to the other end in exactly one hour's time. You know nothing else about the fuses; in particular you don't know how long any segment of a fuse will burn, only that an entire fuse takes one hour. How can you tell when exactly 45 minutes have passed?

ANSWER. This is possible, but it takes two good ideas. The first good idea is to light both ends of a fuse at once. It will burn in a half hour. The second good idea is to light one end of the other fuse at the same time, and its other end after a half hour (as measured by the first fuse). It will finish in 45 minutes: a half hour at normal speed and 15 minutes from both ends. The two fuses need not be identical.

The best, winning entries also consider other time periods. Joe Shipman points out that with a single fuse you can time any fraction 1/n of an hour, by snipping it into pieces and always keeping n ends burning. Similarly with two fuses, any fraction 1/n + 1/m. John Robertson notes that even without snipping, with two fuses you can time any fraction of the form n/4 [except perhaps 1/4?], and with three fuses any fraction of the form n/8, except perhaps 5/8? Elliot Kearsley warns against trying these experiments in your living room. Len VanWyk suggests that with enough fuses, you could trade them in for a watch.

Todd Feitelson points out that two basic theorems of calculus, the Intermediate Value Theorem and the Mean Value Theorem, have applications to fuses: "Each fuse burns continuously from time T=0 to T=60, which means there is a point on each fuse where T=30. This is a 'Time Center' rather than a 'Geometric Center,' but the Intermediate Value Theorem guarantees that it exists. . . . These poorly made fuses which burn irregularly take exactly one hour to burn. The Mean Value Theorem guarantees that there is a point on each fuse where the rate of burning is exactly 1/60 of the total length of the fuse per minute."